Hey JP - to my knowledge there are no dog parks on Siesta Key. I may get in trouble for saying this, but although dogs are not allowed on the beaches, in the evenings when there are less people we often see dogs. And I for one think its awesome. But I dont think its officially allowed.
Not any real running paths either, but we do have sidewalks and many, many quiet streets. If you were to live on Siesta, but want a true running path, I would suggest going a few miles over the South Bridge to the new Legacy Trail, a beautiful 17 mile (give or take) wooded running and bike path with entrances/exits every mile or so. There is always running on the beach!
As for other recreation areas, Siesta Key Beach is the most well known of course. While crowded, its great for families because there are restrooms, concessions, and showers. There are playgrounds, picnic areas, a beach volleyball court, and the Siesta Key drumcircle. If you dont have kids with you (so you dont have to keep going back to the bathroom) I would park at the very north end of the lot, then walk north from there until you find your deserted stretch of beach.
A less popular beach area is Turtle Beach Park at the south end of the Key. The beach is much smaller, and the parking lot much smaller to traverse as well. But there are great quiet areas along the inlets, with a shaded playground and picnic area. There is a boat ramp where you can also launch kayaks and canoes, jetskis into the Bay - dolphins, mangroves, quiet water. Its beautiful.
If you like fishing, there is a great little park directly at the north causeway bridge, in the south side. You can park, watch the boats and guys fishing and crabbing. The birds know what to expect and you will often share a bench with a heron or ibis as they wait for the fish haul. Walk under the bridge to see a great view of the Sarasota skyline and John Ringling bridge (heres the view - http://www.SeeSarasotaLive.com
Finally, a great little secret - walk out on Avenida Messina, just north of the village to the beach - its rocky, and wilder looking, and has a whole different feel than the expansive white beaches.
The magic of the Key is in exploring it. There may not be many "official" parks but nature is at every turn. I hope this helps.